The lengths of components have a normal distribution, with a mean of \(7 \mathrm{~cm}\) and a standard deviation of \(0.03 \mathrm{~cm}\). Calculate the probability that a component chosen at random has a length (a) between \(6.95 \mathrm{~cm}\) and \(7.02 \mathrm{~cm}\) (b) more than \(7.05 \mathrm{~cm}\) (c) less than \(6.96 \mathrm{~cm}\) (d) between \(6.95 \mathrm{~cm}\) and \(6.99 \mathrm{~cm}\).

The Z-score is a powerful statistical tool used to determine the position of a specific value in relation to the mean of a group of values, when the data is normally distributed. In essence, the Z-score formula transforms any given value into standard deviations away from the mean.

The formula is expressed as:\[\begin{equation}Z = \frac{X - \text{{\textmu}}}{\text{{\textsigma}}}\text{{\textend{equation}}}where \(Z\) represents the Z-score, \(X\) is the observed value, \(\textmu\) is the mean, and \(\text{{\textsigma}}\) is the standard deviation.

It's essential to grasp that a Z-score of 0 indicates the observed value is exactly at the mean, a positive Z-score means the value is above the mean, and a negative Z-score denotes a value below the mean. This concept is particularly instrumental when comparing different data points within the same distribution or across different distributions with diverse scales.

For instance, in the exercise provided, the Z-scores calculated for various component lengths allow us to compare their positions relative to the average length of components. Understanding Z-scores enriches our comprehension of probability and how rare or common specific measurements are within a given context.

The standard normal distribution, also known as the Z-distribution, is a special case of the normal distribution. It has a mean (\(\textmu\)) of 0 and a standard deviation (\(\text{{\textsigma}}\)) of 1. This distribution is portrayed through a bell-shaped curve where the majority of the data points lie close to the mean, decreasing in frequency as they deviate further away.

The beauty of the standard normal distribution lies in its ability to standardize different normal distributions, allowing for a universal approach to probability calculation. By converting values to Z-scores, we can use the standard normal distribution table, or Z-table, to find the probabilities associated with these scores.

Understanding the standard normal distribution is crucial since it enhances our ability to gauge the likelihood of various outcomes. By referring to Z-tables or using statistical software, we can instantly ascertain the probabilities of Z-scores, streamlining the probability calculation process.

Probability calculation is the cornerstone of understanding how likely any given event is to occur. In the context of normal distribution, we employ the Z-score and the standard normal distribution to ascertain this likelihood.

To calculate the probability of a data point falling within a specific range, we determine the Z-scores at each endpoint of the range. We then look up these Z-scores in the Z-table to find their corresponding probabilities. The probability of the data point falling in the stipulated range is the difference between these probabilities for a range or one minus the probability for a single cutoff when looking for the probability of being greater than a given value.

In the textbook example, we calculated the probability of a component's length being within certain ranges by finding the relevant Z-scores and employing the Z-table. These calculations are key to making informed decisions in various fields, from quality control in manufacturing to determining risk factors in finance. Fluent probability calculation skills prove indispensable in academic pursuits and in practical, real-world scenario analysis.